One of the major problems facing heart transplant recipients is acute and chronic rejection of their engrafted heart, with resultant loss of graft function. No currently available immunosuppressive regimen can completely prevent rejection without significant side effects. Although the actual mechanism of chronic rejection remains unresolved, it is clear that eventually blockage of the blood vessels of the graft occurs as the lining of the blood vessels thicken. Higher doses of the conventional immunosuppressive drug, cyclosporin, have been used in an attempt to overcome this problem, but high doses of cyclosporin lead to an increased risk of opportunistic infections, kidney toxicity, and susceptibility to malignancies. This study is designed to evaluate a different immunosuppressive agent that could act synergistically with cyclosporin, and therefore permit lower doses of cyclosporin to be used. That drug is rapamycin, a product of a mold that has been found to have immunosuppressive properties in laboratory experiments and in intact animals. Its mechanism of action appears to be different from that of cyclosporin. Rapamycin appears to inhibit growth factor-dependent proliferation of blood cells that have the capacity to secrete substances injurious to the graft. This novel mechanism of action makes rapamycin an attractive agent to use in combination with cyclosporin. The purpose of this study is to determine the incidence of chronic rejection in patients receiving rapamycin in comparison to those receiving azathioprine (as measured by intracoronary vascular ultrasound to determine the degree of thickening of the coronary arteries).